\(\int e^{-\text {sech}^{-1}(a x)} x \, dx\) [79]

Optimal result

Integrand size = 10, antiderivative size = 94 \[ \int e^{-\text {sech}^{-1}(a x)} x \, dx=\frac {(1+a x)^2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}{4 a^2}+\frac {(1+a x) \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^2}+\frac {\arctan \left (\sqrt {\frac {1-a x}{1+a x}}\right )}{a^2} \]

[Out]

arctan(((-a*x+1)/(a*x+1))^(1/2))/a^2+1/4*(a*x+1)^2*(1-((-a*x+1)/(a*x+1))^(1/2))^2/a^2+1/2*(a*x+1)*(1+((-a*x+1)
/(a*x+1))^(1/2))/a^2

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6472, 833, 653, 209} \[ \int e^{-\text {sech}^{-1}(a x)} x \, dx=\frac {\arctan \left (\sqrt {\frac {1-a x}{a x+1}}\right )}{a^2}+\frac {(a x+1)^2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}{4 a^2}+\frac {(a x+1) \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}{2 a^2} \]

[In]

Int[x/E^ArcSech[a*x],x]

[Out]

((1 + a*x)^2*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2)/(4*a^2) + ((1 + a*x)*(1 + Sqrt[(1 - a*x)/(1 + a*x)]))/(2*a^2)
+ ArcTan[Sqrt[(1 - a*x)/(1 + a*x)]]/a^2

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(p + 1), x] + Dist[d*((2*p + 3)/(2*a*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
- 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 6472

Int[E^(ArcSech[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[(1 - u)/(1 + u)] + (1/u)*Sqrt[(1 - u)/(
1 + u)])^n, x] /; FreeQ[m, x] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x}{\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}} \, dx \\ & = -\frac {4 \text {Subst}\left (\int \frac {(-1+x)^2 x}{\left (1+x^2\right )^3} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^2} \\ & = \frac {(1+a x)^2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}{4 a^2}-\frac {\text {Subst}\left (\int \frac {-2+2 x}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^2} \\ & = \frac {(1+a x)^2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}{4 a^2}+\frac {(1+a x) \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^2}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^2} \\ & = \frac {(1+a x)^2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}{4 a^2}+\frac {(1+a x) \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^2}+\frac {\arctan \left (\sqrt {\frac {1-a x}{1+a x}}\right )}{a^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.80 \[ \int e^{-\text {sech}^{-1}(a x)} x \, dx=-\frac {-2 a x+a x \sqrt {\frac {1-a x}{1+a x}} (1+a x)+i \log \left (-2 i a x+2 \sqrt {\frac {1-a x}{1+a x}} (1+a x)\right )}{2 a^2} \]

[In]

Integrate[x/E^ArcSech[a*x],x]

[Out]

-1/2*(-2*a*x + a*x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x) + I*Log[(-2*I)*a*x + 2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a
*x)])/a^2

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.88 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00

method	result	size
default	\(a \left (\frac {x}{a^{2}}-\frac {\sqrt {\frac {a x +1}{a x}}\, x \sqrt {-\frac {a x -1}{a x}}\, \left (\sqrt {-a^{2} x^{2}+1}\, x \,\operatorname {csgn}\left (a \right ) a +\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a x}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \operatorname {csgn}\left (a \right )}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}\right )\)	\(94\)

[In]

int(x/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

a*(x/a^2-1/2/a^2*((a*x+1)/a/x)^(1/2)*x*(-(a*x-1)/a/x)^(1/2)*((-a^2*x^2+1)^(1/2)*x*csgn(a)*a+arctan(csgn(a)*a*x
/(-a^2*x^2+1)^(1/2)))/(-a^2*x^2+1)^(1/2)*csgn(a))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.84 \[ \int e^{-\text {sech}^{-1}(a x)} x \, dx=-\frac {a^{2} x^{2} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 2 \, a x - \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right )}{2 \, a^{2}} \]

[In]

integrate(x/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)),x, algorithm="fricas")

[Out]

-1/2*(a^2*x^2*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 2*a*x - arctan(sqrt((a*x + 1)/(a*x))*sqrt(-(a*x -
1)/(a*x))))/a^2

Sympy [F]

\[ \int e^{-\text {sech}^{-1}(a x)} x \, dx=a \int \frac {x^{2}}{a x \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + 1}\, dx \]

[In]

integrate(x/(1/a/x+(1/a/x-1)**(1/2)*(1+1/a/x)**(1/2)),x)

[Out]

a*Integral(x**2/(a*x*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)) + 1), x)

Maxima [F]

\[ \int e^{-\text {sech}^{-1}(a x)} x \, dx=\int { \frac {x}{\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}} \,d x } \]

[In]

integrate(x/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)),x, algorithm="maxima")

[Out]

integrate(x/(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x)), x)

Giac [F]

\[ \int e^{-\text {sech}^{-1}(a x)} x \, dx=\int { \frac {x}{\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}} \,d x } \]

[In]

integrate(x/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)),x, algorithm="giac")

[Out]

integrate(x/(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x)), x)

Mupad [B] (verification not implemented)

Time = 13.36 (sec) , antiderivative size = 407, normalized size of antiderivative = 4.33 \[ \int e^{-\text {sech}^{-1}(a x)} x \, dx=\frac {x}{a}-\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\,1{}\mathrm {i}}{2\,a^2}-\frac {\frac {1{}\mathrm {i}}{32\,a^2}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{16\,a^2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{32\,a^2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}}{\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}}-\frac {\left (\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1\right )-\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\right )\,1{}\mathrm {i}}{a^2}+\frac {\ln \left (\frac {2\,a\,\sqrt {\frac {a+\frac {1}{x}}{a}}-\frac {2}{x}+a\,\sqrt {-\frac {a-\frac {1}{x}}{a}}\,2{}\mathrm {i}}{2\,a+\frac {1}{x}-2\,a\,\sqrt {\frac {a+\frac {1}{x}}{a}}}\right )\,1{}\mathrm {i}}{2\,a^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{32\,a^2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2} \]

[In]

int(x/((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x)),x)

[Out]

x/a - (log(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1))*1i)/(2*a^2) - (1i/(32*a^2) + (((1/(a*x) - 1)^
(1/2) - 1i)^2*1i)/(16*a^2*((1/(a*x) + 1)^(1/2) - 1)^2) - (((1/(a*x) - 1)^(1/2) - 1i)^4*15i)/(32*a^2*((1/(a*x)
+ 1)^(1/2) - 1)^4))/(((1/(a*x) - 1)^(1/2) - 1i)^2/((1/(a*x) + 1)^(1/2) - 1)^2 + (2*((1/(a*x) - 1)^(1/2) - 1i)^
4)/((1/(a*x) + 1)^(1/2) - 1)^4 + ((1/(a*x) - 1)^(1/2) - 1i)^6/((1/(a*x) + 1)^(1/2) - 1)^6) - ((log(((1/(a*x) -
1)^(1/2) - 1i)^2/((1/(a*x) + 1)^(1/2) - 1)^2 + 1) - log(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1))
)*1i)/a^2 + (log((a*(-(a - 1/x)/a)^(1/2)*2i - 2/x + 2*a*((a + 1/x)/a)^(1/2))/(2*a + 1/x - 2*a*((a + 1/x)/a)^(1
/2)))*1i)/(2*a^2) - (((1/(a*x) - 1)^(1/2) - 1i)^2*1i)/(32*a^2*((1/(a*x) + 1)^(1/2) - 1)^2)